I recently started reading functional analysis on my own and have come about dual spaces and cannot get an intuitive understanding about them. This is where my intuition breaks down while understanding duals of $L^p[0,1]$ spaces for $1\le p<\infty$:
Dual of $L^p[0,1]$ is $L^q[0,1]$ where $1/p+1/q=1$. Here, a dual space consists of 'Linear functionals' (say $l(.)$) that map elements in $L^p[0,1]$ to $R$, whereas $L^q[0,1]$ contains functions (say $f(.)$) that map real numbers to real numbers. In otherwords, the map $l(.)$ works on functions and the map $f(.)$ works on real numbers. How can the linear functionals (dual space) be the same as an elements in $L^q[0,1]$.
I do not have a rigorous mathematics background, If this question is vague or trivial please provide references of better sources which contain simple examples.